# Full text of "Treatise On Analysis Vol-Ii"

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```324       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

the linear forms /H-> I      f(t)e1^ dt, where f is real. (A character corresponds to the

class of a A-measurable function g such that g(s + /) = g(s) + g(t) almost everywhere
with respect to the Lebesgue measure A ® A on R2. Deduce first that g is continuous,
by observing that if #1 =#*p is a regularization of gt then #i(y+ t) = g(s)gi(t).
Deduce then that#(0 == e^\ with £ e C, and show that £ must be pure imaginary.)

7.   Let jtf be the complex vector space of A-measurable functions x on [0, -f- co [ such that
||jf|| = |    |*(/)| sh It dt < -f oo.

(a)    Show that, for all x, y in J/, the function

x(s)y(t) ds

belongs to s£ (use the Lebesgue-Fubini theorem). The mapping (xty)t-+xOy
defines on J/ the structure of a commutative algebra. Let rf be the subspace of
A-negligible functions; then A= «^/^T is a commutative Banach algebra without
unit element.

(b)    Every continuous linear form on A is of the type

x(t)a>(t) dt

where <D is A-measurable and such that |co(OI ^ M sh 2t almost everywhere, M being
a suitable positive constant. Deduce that every character x of the algebra A obtained
by adjoining a unit element to A, such that x IA is not identically zero, corresponds
to a function a> such that

Jl«-s|

almost everywhere with respect to Lebesgue measure on [0, -f oo [ x [0, -f oo [. Show
that, by replacing a* by an equivalent function, we may suppose that w is continuous
and indefinitely differentiable and satisfies the relations co(0) = 0 and

whenever s < t. Deduce that a>(t) = 2(sin pt)/p where p is a complex number, and that
the condition |cu(f)| ^ Msh2r imposes the restriction |«/yo| ^ 2. The complement
in X(A) of the character which vanishes on A can be identified with the orbit space of
the group consisting of the identity and the symmetry £ h-> —• £ acting on the locally
compact subspace of C consisting of the complex numbers p with \Jp\ :g 2.

Let X be a compact space and B a Banach subalgebra of ^C(X) (with respect to the
induced norm) which contains the unit element and separates the points of X. Then
the compact space X can be canonically identified with a closed subset of X(B)
(Problem 1).

For every function /eB,  the  function ^B/ extends / and  is  such  that
||/||= sup  |(^B/Xx)l: *n other words,   ^B is an isometric isomorphism of B

onto a Banach subalgebra of #C(X(B)) (cf. Problem 1). Show that if &f^ 0 then
^0. (Consider the function e~f e B.)ll t e X, then the inverse x"1 of x in ^C(X) belongs to
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